Let `A_n` be the area bounded by the curve `y=(tanx)^n` and the lines `x=0,y=0,` and `x=pi/4dot` Prove that for `n >2,A_n+A_(n-2)=1/(n-1)` and deduce `1/(2n+2)

Options

A. `(1)/(n+1)`
B. `(1)/(n)`
C. `(1)/(n-1)`
D. None of these

Answer

C

Explanation

`A_(n) = underset(0)overset(pi//4)inttan^(n)x.dx = underset(0)overset(pi//4)inttan^(n-2)x(sec^(2)x-1)dx`
`=[(tan^(n-1))/(n-1)]_(0)^(pi//4)-"in"-2`
`rArr A_(n) + A_(n-2)=1/(n-1)`